The Proof of the Isosceles Triangle Theorem

An isosceles triangle is a triangle whose two sides are congruent. In the figure below, ABC is an isosceles triangle and \overline{AB} \cong \overline{BC}.

The Isosceles Triangle Theorem states that the angles opposite to the two congruent sides of an isosceles triangle are congruent. In the figure above, the theorem states that since AB \cong BC, \angle A \cong \angle C.

The proof to this theorem uses the SSS triangle congruence. The SSS Triangle Congruence Theorem states that if the three corresponding sides of two triangles are congruent, then the two triangles are congruent.

 

isosceles-triangle

Isosceles Triangle Theorem

The angles opposite the two congruent sides of an isosceles triangle are congruent.

Given

Isosceles \triangle ABC with \overline {AB} \cong \overline{BC}.

What to Show

\angle A \cong \angle C

Proof

Let D be the midpoint of \overline{AC}.

Connect \overline{BD}.

isosceles triangle

 

Now from the given and the definition of isosceles triangle, \overline{AB} \cong \overline{BC} (S).

Also, since D is the midpoint of \overline{AC}, \overline{AD} \cong \overline{CD}. (S)

In addition, by Reflexive Property, \overline{BD} \cong \overline{BD}. (S)

By the SSS congruence theorem, \triangle BDA \cong \triangle BDC.

Since corresponding parts of congruent triangles are congruent, \angle A \cong \angle C and we are done.

 

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